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Math Help - Integration Problem (Substitution)

  1. #1
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    Integration Problem (Substitution)

    Here's the problem: indefinite integral of (-3x^2)/sqrt((t^3)-16)
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Integration Problem (Substitution)

    Are we integrating with respect to x or t? I suspect you mean for the two variables to be the same, and if so, let u be the value under the radical, and you will find you can easily get du as part of the integral.
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  3. #3
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    Re: Integration Problem (Substitution)

    Yes, I meant them to be the same, oops!

    I got the answer to be sqrt((t^3)+16)/2 + C, but I don't think that's correct...
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: Integration Problem (Substitution)

    If you are unsure of your result, use differentiation to check your result:

    \frac{d}{dx}\left(\frac{\sqrt{t^3+16}}{2}+C \right)=\frac{3t^2}{4\sqrt{t^3+16}}\ne\frac{-3t^2}{\sqrt{t^3+16}}

    We can see it is close (essentially the wrong constant factor of the radical), but not quite right. I would let:

    u=t^3+16\,\therefore\,du=3t^2\,dt and so we have:

    -\int u^{-\frac{1}{2}}\,du=-2u^{\frac{1}{2}}+C=-2\sqrt{t^3+16}+C

    Now, checking by differentiation, we find:

    \frac{d}{dx}\left(-2\sqrt{t^3+16}+C \right)=\frac{-3t^2}{2\sqrt{t^3+16}}

    The derivative of the anti-derivative is the original integrand, so our result is correct.
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